Self-lensing flares from black hole binaries: General-relativistic ray tracing of black hole binaries

The self-lensing of a massive black hole binary (MBHB), which occurs when the two BHs are aligned close to the line of sight, is expected to produce periodic, short-duration flares. Here we study the shapes of self-lensing flares (SLFs) via general-relativistic ray tracing in a superimposed binary BH metric, in which the emission is generated by geometrically thin accretion flows around each component. The suite of models covers eccentric binary orbits, black hole spins, unequal mass binaries, and different emission model geometries. We explore the above parameter space and report how the light curves change as a function of, e.g., binary separation, inclination, and eccentricity. We also compare our light curves to those in the microlensing approximation, and show how strong deflections, as well as time-delay effects, change the size and shape of the SLF. If gravitational waves (GWs) from the inspiraling MBHB are observed by LISA, SLFs can help securely identify the source and localizing it on the sky, and to constrain the graviton mass by comparing the phasing of the SLFs and the GWs. Additionally, when these systems are viewed edge-on the SLF shows a distinct dip that can be directly correlated with the BH shadow size. This opens a new way to measure BH shadow sizes in systems that are unresolvable by current VLBI facilities.

Figure 1

Illustration of the model setup. The binary’s center of mass is at the origin, and the observer is located in the $x\text{−}z$ plane. The binary’s orbital plane is tilted with respect to the observer’s line of sight by the angle $i_{\text{orbit}}$. The BH spin axes and the orbital angular momentum vectors of the minidisks are kept parallel to the $z$-axis and are misaligned with the binary’s orbital plane by the angle $I$. The node angle $\mathrm{\Omega }$ specifies the orientation of the semimajor axis of the elliptical binary orbit within the orbital plane; rotating the binary while keeping the observer’s position fixed will change the moment during the orbit when the two BHs are aligned with the line of sight.

Figure 2

Example of a ray-traced image in one of our models, illustrating the adaptive grid of the camera pixelization. White lines indicate blocks of equal resolution, each containing ${25}^2\mathrm{pixels}$. Higher resolution blocks are triggered only when a block overlaps with the source.

Figure 3

Black-body spectra for a binary mass of $M={10}^7{M}_{\odot }$, and mass ratio $q=1$ at various radii in the disk. The temperature is normalized so that the spectra resemble those found in hydrodynamical simulations [5].

Figure 4

Light curve and images in the fiducial model. The top three panels show images of the binary when they have the largest projected separation on the sky (top panel), and at the rise and peak of the lensing event (two middle panels). Blue colors indicate an approaching BH, orange the receding BH. Bottom panel: the combined light curve of the binary, with numbers indicating the moments shown in the upper three panels.

Figure 5

Various light curves showing the impact of different physical effects. The top panel shows the difference between microlensing models, assuming either point source, a source with a Gaussian surface brightness distribution, or the actual source BH image in our model. The second panel shows a comparison between microlensing and GR-generated light curves. The third panel shows the effect of relativistic Doppler boosting. The bottom panel takes all effects into account by also including time-delays.

Figure 6

Geodesics in gray as an illustration of strong and weak deflections. Orange disks indicate the minidisks around each BH and black circles mark the event horizons. The observer is to the right, so that the emission from the left-side BH is lensed by the right-side BH. The rays originating from the source at positive $y$ and passing the lens at negative $y$ suffer strong deflections and form a secondary image whose brightness is not predicted accurately by microlensing.

Figure 7

Light curves showing the inner-radius dependence by contrasting the fiducial model M0 (with the inner radius at the event horizon) and its variant M1 (with the inner radius moved out to ISCO). As the inner radius increases, there is a larger central gap in the accretion flow, extending outside the horizon. This enhances the amplitude of the flare, with a wider and broader “dip” (top two panels). Additionally, the ISCO is outside the photon sphere, resulting in a photon ring in the image plane of the source. The photon ring adds an extra minor increase to the light curve on either side of the central dip (bottom panel).

Figure 8

Light curves showing the dependence on the observer’s viewing angle. The fiducial model M0 has an inclination of 90° (edge-on), and the models M2a—f have smaller inclinations of 89°, 88°, 87°, 86°, 85°, and 80° respectively. As the inclination decreases, the separation on the image plane between the source and the lens increases, resulting in a lower overall amplification, as expected from microlensing-based models [22]. While the overall flare remains visible even for disks misaligned by 10°, the central dip disappears once the BH shadow’s projected offset from the lens is too large (that is, in the models M2d,e,f which are more than $\sim 3°$ from edge on).

Figure 9

Light curves showing the dependence on the separation of the two BHs. The models M3a—e have separations of 200, 300, 400, 500, and $1000R_{\mathrm{g}}$, compared to $100R_{\mathrm{g}}$ in the fiducial model. As the separation increases, the angular size of the source becomes smaller, resulting in a larger amplification since a larger fraction of the source falls inside the Einstein angle, as well as a narrower width in phase, since the source spends a smaller fraction of the total orbit behind the lens; however, since the period does grow, the width becomes large in physical time.

Figure 10

Light curves showing the eccentricity dependence. The fiducial model is circular, while models M4a—c have increasing eccentricities of $e=0.3$, 0.6, and 0.9, respectively. The pericenter distance is kept constant in all four models. The bottom panel shows the orbits in the $x\text{−}y$ plane, with the observer to the right at $y=0$. The more eccentric the binary, the larger the separation at apoapsis, and the smaller the angular size of the source; this results in a higher amplification of the flux, similarly as for models M2a—e in Fig. 9. At periapsis, the orbital velocity increases with $e$, and therefore the source spends a smaller fraction of the orbit directly behind the lens making the flares narrower.

Figure 11

Light curves showing the node angle dependence. All curves have eccentricities of $e=0.9$, as in model M4c, but models M5a—c change the nodal angle ($\mathrm{\Omega }=0$ in model M4c) to $\mathrm{\Omega }=30$, 60 and 90°, respectively. The bottom panel shows the orbits in the $x\text{−}y$ plane, with the observer to the right at $y=0$. As the nodal angle changes, the BHs align at different phases along the orbit and align with the minor axis in the case of model M5c. Since the lensing events happen at the closest approach here, the spacing becomes nonuniform compared to M3a. Since the separation decreases, the amplification drops, and the width narrows due to higher velocities.

Figure 12

Light curves showing the mass-ratio dependence. Models M6a and M6b differ from the fiducial model ($q=1$) by having $q=0.1$, and $q=0.3$, respectively. As the mass ratio decreases, the angular size of the secondary BH’s disk decreases, which results in a higher amplification factor when it is being lensed. Simultaneously, the ratio between the primary source size and the Einstein radius of the secondary as the lens becomes larger, resulting in lower amplification and widening the flare profile when the primary BH is lensed.

Figure 13

Light curves showing the spin magnitude dependence. The fiducial model has zero spin ($a=0$) while models M7a—b have spins of $a=0.9$ and 0.95, respectively. The spin dependence only has a relatively small effect on the asymmetry of the double-peaked structure, since it only modifies the emission and the space-time metric within a few gravitational radii of the BHs.

Figure 14

Light curves showing the spin orientation dependence. The BH spins and their minidisks in models M8a—b differ in their inclination with respect to the line of sight. In the fiducial model, both minidisks have $i_{\mathrm{disk}}=90°$ (edge-on), whereas models M8a—b have $i_{\mathrm{disk}}=0°$ (face-on) and $i_{\mathrm{disk}}=45°$, respectively. For nearly face-on minidisks, a new effect appears, as the foreground disk can block the light from the lensed BH—this physical occultation can erase the flare or replace it with a transitlike depression.

Figure 15

Light curves showing the dependence on the spectral shape, via changing the BH masses. Models M9a—b have a BH mass of ${10}^5{M}_{\odot }$ and ${10}^9{M}_{\odot }$, respectively, compared to ${10}^7{M}_{\odot }$ in the fiducial model. This changes the spectral slope in the observed X-ray band (between 2.5 an 10 keV) to $\alpha =1.1$, 1.7, and $-0.8$, respectively. The spectral slope only modestly affects the overall flare amplitude. We expect it to have a larger impact on the overall Doppler modulation during the orbit in the case of an unequal-mass BH, where the Doppler effects from the two BHs do not nearly cancel.

Figure 16

Light curves demonstrating the impact of optical depth. Model M10 is identical to the fiducial model M0, except the minidisks are assumed artificially to be optically thin, while in the fiducial model M0 they are optically thick. The sharp features seen for M10 arise from the photon rings. These can only form in the optically thin case and concentrate the emission to a small solid angle, allowing stronger amplification when parts of the bright ring are behind the lens.